Faculty of Science and Technology
MASTER’S THESIS
Springsemester, 2018
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Stavanger, June 29, 2018 Date/year Study program/Specialization:
MSc in Offshore Technology / Marine and Subsea Technology
Marek Jan Janocha
Prof. Muk Chen Ong
30 Thesistitle:
CFD Simulations of Vortex-Induced Vibrations of a Subsea Pipeline Near a Horizontal Plane Wall
vortex shedding, laminar flow, turbulent flow, near-wall, CFD, URANS, OpenFOAM, vortex induced vibration, circular cylinder, piggyback Programme coordinator:
Prof. Muk Chen Ong
143 24
CFD SIMULATIONS
OF VORTEX-INDUCED VIBRATIONS OF A SUBSEA PIPELINE NEAR
A HORIZONTAL PLANE WALL
Author:
Marek Jan Janocha
Supervisor:
Prof. Muk Chen Ong University of Stavanger
UNIVERSITY OF STAVANGER Faculty of Science and Technology
Department of Mechanical and Structural Engineering and Materials Science Master of Science Thesis, Spring 2018
To my Family
ABSTRACT
Subsea pipelines, when exposed to free spans, can experience vortex-induced vibrations.
This phenomenon was described as a resonance condition occurring when the vortex shedding frequency and the natural frequency of a structure approach common oscilla- tion frequency. Fatigue life of the pipeline can be adversely affected by a high amplitude oscillations attributed to the vortex-induced vibrations. A numerical study has been per- formed on the effects of wall proximity on the vortex shedding of an elastically mounted circular cylinder. In addition, the study was extended to investigate the influence of a sec- ond cylinder with a smaller diameter rigidly coupled with the large cylinder. Such config- urations can be regarded as a model of a subsea pipeline or, in case of coupled cylinders, a subsea piggyback pipeline in a free span situation. A series of two dimensional, numer- ical studies using open source CFD code OPENFOAM has been performed. Simulations were performed in two flow regimes, a laminar vortex street regime at Reynolds number Re= 200and an upper transition regime atRe= 3.6×106. A range of reduced velocities covering a frequency lock-in phenomenon was investigated. Hydrodynamic forces and response amplitudes were mapped with respect to the reduced velocity. Furthermore, a study of the phase differences between the hydrodynamic forces and cylinder displace- ments was conducted. The motions of the cylinder were recorded and presented on the trajectory plots. The frequency components of hydrodynamic forces and displacements were analyzed with the FFT algorithm in the frequency domain. In order to gain insight into the effects of the shear layers interaction in the area around the oscillating cylinder, flow visualizations of the numerical simulations were analyzed.
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ACKNOWLEDGMENTS
I would like to thank Prof. Muk Chen Ong for giving me the possibility to work under his supervision during my studies at the University of Stavanger. Moreover, I would like to sincerely thank him not only for his helpful and inspiring guidance, but also for providing me the opportunity to develop my interest in Computational Fluid Dynamics.
I would also like to thank University of Stavanger, Department of Mechanical and Struc- tural Engineering and Materials Science for providing all the resources necessary to com- plete presented work.
This study was supported in part with computational resources provided by the Norwe- gian Metacenter for Computational Science (NOTUR) under Project No. NN9372K. This support is greatly acknowledged.
In closing, I like to thank my parents, my family and friends for their aid, and support, and Dominika for her endless love and encouragement.
M. J. JANOCHA
Stavanger, Norway June, 2018
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Contents
Abstract v
Acknowledgments vii
List of Figures xiii
List of Tables xvii
Acronyms xix
List of Symbols xxi
1 Introduction 1
1.1 Background and Motivation 1
1.2 Problem Definition and Objectives of the Thesis 2
1.3 Outline of the Thesis 3
1.4 Previous Work 4
1.5 Summary 8
References 9
2 Flow Around Circular Cylinder and Vortex-Induced Vibration 13
2.1 Flow Around Immersed Cylinders 13
2.1.1 Flow Regimes 14
2.2 Turbulence 15
2.3 Boundary Layer Concept 17
2.3.1 Laminar Boundary Layer 17
2.3.2 Turbulent Boundary Layer 19
ix
2.3.3 Plus Units 20
2.3.4 Law-of-the-wall 20
2.4 Vortex Shedding 21
2.4.1 Mechanism of Vortex Shedding 21
2.4.2 Vortex Shedding Patterns 22
2.5 Hydrodynamic Forces 23
2.5.1 Forces on a Cylinder in Steady Current 23
2.5.2 Drag and Lift Coefficients 24
2.5.3 Pressure Coefficient and Skin-friction Coefficient 25
2.6 Dynamic Equations of Motion 25
2.6.1 Solutions of a Viscous Damped Vibration Equation 25
2.6.2 Dynamics of 2-Dof Systems in Fluid Flow 26
2.7 Vortex-Induced Vibrations 27
2.7.1 Influencing Parameters 27
2.7.2 Strouhal Number 28
2.7.3 Mass Ratio 29
2.7.4 Reduced Velocity 29
2.7.5 Added Mass 29
2.7.6 Frequency Lock-in 30
2.7.7 Response Amplitude 31
2.7.8 Effect of Wall Proximity 32
2.8 Summary 33
References 35
3 Computational Fluid Dynamics 37
3.1 Introduction 37
3.2 OpenFOAM 38
3.2.1 Case Directory Structure 38
3.3 Governing Equations 39
3.4 The Finite Volume Method 40
3.4.1 Spatial Discretisation 40
3.4.2 Temporal Discretisation 42
3.4.3 Equation Discretisation 43
3.4.4 Pressure - Velocity Coupling 43
3.5 Turbulence Modeling 45
3.5.1 k−ωSST Turbulence Model 46
References 48
4 Two degree-of-freedom near wall VIV in Laminar Vortex Street Regime 51
4.1 Pre-processing 51
4.2 Model Description 52
4.3 Convergence Studies 53
4.4 Results and Discussion 58
4.4.1 Hydrodynamic Forces 59
CONTENTS xi
4.4.2 Response Amplitudes 60
4.4.3 Phase Difference of Forces and Responses 62
4.4.4 Frequency Response 63
4.4.5 Motion Trajectories 64
4.4.6 Flow Field Characteristics 65
4.4.7 Vibration Frequency Characteristics 69
4.4.8 Vortex Shedding Modes 70
4.5 Summary 72
References 75
5 Paper I: VIV of two rigid cylinders with uneven diameters near the horizontal
plane wall at low Re 77
6 Two degree-of-freedom near wall VIV in Upper Transition Regime 111
6.1 Computational Domain and Boundary Conditions 111
6.1.1 Convergence Studies 113
6.1.2 Model Validation 117
6.2 Results and Discussion 118
6.2.1 Response Amplitudes 119
6.2.2 Hydrodynamic Forces 121
6.2.3 Phase Pictures and Motion Trajectories 122
6.2.4 Flow Field Charactersitics 125
6.3 Summary 135
References 137
7 Conclusions and Recommendations 139
7.1 Conclusions 140
7.2 Recommendations for Future Work 143
A Results of simulations in
laminar vortex street regime 145
A.1 Time histories ofCD,CL,X/DandY /Dand power spectral analysis 146 B Results of simulations in
upper transition regime 153
B.1 Time histories of displacementsx/D,y/D, and force coefficientsCD,CL 154
B.2 FFT ofy/DandCLat investigatedUr 161
B.3 Phase picturesφCL−Y andX−Y trajectory plots at investigatedUr 165
LIST OF FIGURES
1.1 Free spanning subsea pipeline. 2
1.2 Example of a piggyback pipeline and its 2D model generalization. 3 2.1 Regions of disturbed flow around circular cylinder. 14
2.2 Velocity profile in a boundary layer. 17
2.3 Separation of a boundary layer over curved surface. 18 2.4 Formation of the laminar and turbulent boundary layer. 19
2.5 The inner layer of a turbulent boundary layer. 21
2.6 Vortex shedding mechanism. 22
2.7 Pressure distribution and forces during vortex shedding cycle. 24 2.8 Strouhal - Reynolds number relationship for circular cylinders. 28 2.9 Experimental vortex shedding frequencies and oscillation frequencies of
submerged cylinder. 30
2.10 Cross-flow response of a flexibly-mounted circular cylinder subject to steady
current. 31
xiii
2.11 Overview diagram of a low mass-damping type response. 32 2.12 Changes in pressure distribution and stagnation point - cylinder placed in a
proximity of a wall. 33
2.13 Streamlines and stagnation point around the cylinder in freestream and
near-wall configuration. 33
3.1 OpenFOAM simulation case directory structure. 38
3.2 Control Volume example. 41
3.3 Structured and unstructured meshes examples. 42
3.4 Flowchart of PIMPLE algorithm. 44
4.1 Schematic of a computational domain and imposed boundary conditions at
Re= 200. 52
4.2 Mesh topology schematic. 54
4.3 Computational mesh details - mesh B, 70040 cells. 54
4.4 Enlarged views of the mesh B details. 55
4.5 Mesh density convergence plots ofCD andCLrmsatRe= 200. 56 4.6 Time step convergence plots ofCDandCLrmsatRe= 200. 57 4.7 Force coefficients as a function ofUratRe= 200. 60 4.8 Time history of drag and lift coefficients of a vibrating cylinder atUr = 4.1
andRe= 200. 61
4.9 Normalized peak transverse displacement and root-mean-square streamwise
displacement atRe= 200. 61
4.10 Phase difference between transverse displacement and lift force and
streamwise displacement and drag force atRe= 200. 63
4.11 Vibration frequency responses atRe= 200. 64
4.12 Trajectories of near-wall cylinder for different reduced velocities atRe= 200. 65 4.13 Time histories ofCD,CL,X/DandY /Dand vorticity contours atUr= 3. 66 4.14 Time histories ofCD,CL,X/DandY /Dand vorticity contours atUr= 5. 67 4.15 Time histories ofCD,CL,X/DandY /Dand vorticity contours atUr= 8. 68
LIST OF FIGURES xv
4.20 Vortex shedding modes atUr = 3.0,Ur = 3.9andUr = 4.3for near-wall
cylinder atRe= 200. 72
4.21 Vortex shedding modes atUr = 5.0,Ur = 6.0,Ur = 7.0andUr = 8.0for
near-wall cylinder atRe= 200. 73
6.1 Schematic of a computational domain and imposed boundary conditions at
Re= 3.6×106. 112
6.2 Computational mesh details - mesh M3, 76364 cells. 115 6.3 Mean pressure coefficient around the cylinder at Re = 3.6 ×106;
δ/D= 0.48;e/D= 1.0. 118
6.4 Non-dimensional maximum amplitude of transverse vibrationAY,max/Dat Re= 3.6×106as a function of reduced velocityUr. 120 6.5 Non-dimensional root-mean-square amplitude of streamwise vibration
AX,rms/DatRe= 3.6×106as a function of reduced velocityUr. 120 6.6 Mean lift coefficientCLatRe= 3.6×106as a function of reduced velocity
Ur. 121
6.7 Root-mean-square lift coefficientCLrms atRe= 3.6×106as a function of
reduced velocityUr. 122
6.8 Mean drag coefficientCD atRe = 3.6×106 as a function of reduced
velocityUr. 123
6.9 Root-mean-square drag coefficientCDrmsatRe= 3.6×106as a function of
reduced velocityUr. 123
6.11 Phase picture (φCL−Y) ofCLandy/DandX−Y trajectory; single cylinder;
e/D= 2.0;Re= 3.6×106;Ur = 6. 124
6.13 Phase picture (φCL−Y) of CL andy/D andX−Y trajectory; coupled
cylindersα= 0◦;e/D = 2.0;Re= 3.6×106;Ur= 4. 125 6.15 Phase picture (φCL−Y) of CL andy/D andX−Y trajectory; coupled
cylindersα= 0◦;e/D = 2.0;Re= 3.6×106;Ur= 5. 125 6.17 Phase picture (φCL−Y) of CL andy/D andX−Y trajectory; coupled
cylindersα= 90◦;e/D= 2.0;Re= 3.6×106;Ur= 6. 126 6.19 Phase picture (φCL−Y) of CL andy/D andX−Y trajectory; coupled
cylindersα= 180◦;e/D= 2.0;Re= 3.6×106;Ur = 4. 126
6.21 Phase picture (φCL−Y) of CL andy/D andX−Y trajectory; coupled
cylindersα= 180◦;e/D= 2.0;Re= 3.6×106;Ur = 10. 127 6.23 Vorticity contours and streamlines with normalized pressure contours for a
single cylinder atUr = 6andRe= 3.6×106. 128 6.24 Time histories of CD, CL, x/D andy/D; single cylinder at Ur = 6,
Re= 3.6×106. 129
6.26 Vorticity contours and streamlines with normalized pressure contours for a
coupled cylindersα= 0◦atUr = 5andRe= 3.6×106. 130 6.27 Time histories of CD, CL, x/D andy/D; coupled cylinders α = 0◦ at
Ur = 5,Re= 3.6×106. 131
6.29 Vorticity contours and streamlines with normalized pressure contours for a
coupled cylindersα= 90◦atUr = 6andRe= 3.6×106. 132 6.30 Time histories ofCD, CL,x/D andy/D; coupled cylindersα = 90◦ at
Ur = 6,Re= 3.6×106. 133
6.31 Time histories ofCD,CL,x/Dandy/D; coupled cylindersα = 180◦ at
Ur = 10,Re= 3.6×106. 133
6.33 Vorticity contours and streamlines with normalized pressure contours for a
coupled cylindersα= 180◦atUr = 10andRe= 3.6×106. 134
LIST OF TABLES
2.1 Flow regimes around circular cylinder. 16
2.2 Vortex shedding patterns. 23
3.1 Constant values used in thek−ωSST model. 47
4.1 Mesh topology - cell distribution parameters. 55
4.2 Cell distribution of meshes used in the convergence study atRe= 200. 55
4.3 Mesh convergence study results atRe= 200. 56
4.4 Time step independence study results atRe= 200. 57
4.5 Influence of the domain size atRe= 200. 57
4.6 List of simulation cases used in the study atRe= 200. 58 6.1 Summary of cell distribution parameters of the meshes used in the
convergence study atRe= 3.6×106. 114
6.2 Convergence study: Static cylinder, effects of the first cell layer height.
Other parameters of the simulations: Re= 3.6×106,e/D= 1, Mesh M3. 114 6.3 Convergence study: Static cylinder, effects of the grid density. Other
parameters of the simulations: Re= 3.6×106,e/D= 1. 116 xvii
6.4 Convergence study: Vibrating cylinder, effects of the grid density. Other
parameters of the simulations: Re= 3.6×106,e/D= 2,Ur = 6. 116 6.5 Convergence study: Vibrating cylinder, effects of the time step. Other
parameters of the simulations: Re= 3.6×106,e/D= 2,Ur = 6. 117 6.6 Experimental data and numerical results atRe= 3.6×106. 118
ACRONYMS
CF Cross Flow
CFD Computational Fluid Dynamics
CV Control Volume
CWT Continuous Wavelet Transform DNS Direct Numerical Simulation DoF Degree-of-Freedom
FEM Finite Element Method FFT Fast Furrier Transform FVM Finite Volume Method
IL In Line
LES Large Eddy Simulation PIV Iarticle Image Velocimetry RMS Root-Mean-Squared TrW Transition in Wake
URANS Unsteady Reynolds-Averaged Navier-Stokes VIV Vortex-Induced Vibrations
xix
SYMBOLS
ROMAN SYMBOLS
A Amplitude of vibration / control surface c Damping coefficient
CF Skin friction coefficient Cp Pressure coefficient CD Drag coefficient CL Lift coefficient Co Courant number
D Diameter / characteristic length e Gap distance
FD Drag force FL Lift force
xxi
fn Natural frequency of a system fosc Frequency of oscillation
fst Vortex shedding frequency of a body at rest fvac Natural frequency measured in vacuum k Spring stiffness
m Mass m∗ Mass ratio ma Added mass p Pressure
Re Reynolds number St Strouhal number
U∞ Free-stream flow velocity Ur Reduced velocity
u Velocityx-component V Control volume v Velocityy-component w Velocityz-component
x Distance in streamwise direction y Distance in crossflow direction z Distance in spanwise direction
GREEK SYMBOLS
δ Boundary layer thickness
LIST OF SYMBOLS xxiii
µ Dynamic viscosity ν Kinematic viscosity
φ General unknown variable / phase difference ρ Material density
τ Shear stress / dimensionless time ζ Damping factor
CHAPTER 1
INTRODUCTION
1.1 Background and Motivation
Study of the flow patterns around cylindrical structures is relevant in a variety of engi- neering applications. In the offshore industry, many structures can be represented by such geometry. Subsea pipelines are one of the most notable examples. It is estimated that more than 45.000 kilometers of pipelines have been installed in the North Sea since 1966.
From an economic and environmental point of view, subsea pipelines represent an im- portant asset with high safety and reliability requirements imposed by the regulators.
Reynolds number characteristic for flow around pipes at the sea bottom is typically of O(104)−O(107)covering subcritical to transcritical flow regimes. When exposed to fluid flow, free pipeline spans are subjected to dynamic motions induced by currents and/or waves, referred to as vortex-induced vibrations (VIV), which can cause fatigue related failure (Fig. 1.1). Another consequence of VIV is the drag amplification effect which results in enlarged static displacements and tensile forces. It is therefore desirable to de-
1
velop a better understanding of the coupled dynamics of free span pipelines undergoing vibrations in the vicinity of a seabed.
Span shoulder Free span Span shoulder
Alternating vortex shedding
Flow direction
Figure 1.1:Free spanning subsea pipeline (DNV GL, 2017).
In this thesis, numerical simulations are used to study the motion of cylindrical struc- tures undergoing vortex-induced vibrations. A numerical approach offers several ben- efits. Most importantly it allows performing parametric studies where among a large number of influencing parameters one of them can be varied while the others are kept constant. This provides the ability to discern the functional dependencies governing the inherently complex near-wall VIV physics. The second benefit of numerical studies is the ability to go beyond the limitations of experimental facilities, which are often limited with respect to the maximum Reynolds number possible to achieve.
1.2 Problem Definition and Objectives of the Thesis
The problem of a free-span along the pipeline can be modeled by the configuration in which flow is past an elastically mounted circular cylinder with two degree-of-freedom (2-DoF) in proximity to a stationary plane wall. Example of such problem can be seen in Fig. 1.2. Research methodology is based on the numerical study of the flow by using an open source finite volume method (FVM) code Open Field Operation And Manipulation (Weller et al., 2008).
It is a well established Computational Fluid Dynamics (CFD) code used by both academic and commercial organizations. The scope of the thesis is focused on two major studies.
In the first part, the focus is on the flow around cylindrical structures at low Reynolds number. The second part is concerned with the upper transition flow regime. Two- dimensional models of the pipeline cross sections are investigated at Re = 200 in the first part and atRe= 3.6×106in the second part respectively. The results obtained from
OUTLINE OF THE THESIS 3
(a) (b)
Figure 1.2:Example of a piggyback pipeline (a) (Subenesol.co.uk, 2018) and 2D model generalization (b).
CFD study are validated against other numerical and experimental studies. The main objectives established in the present thesis apply to both parts and are listed as follows:
Research objective: The formation and development of vortices around an oscillating cylinder in near-wall configuration and their interaction with the bottom boundary layer shall be investigated.
Research objective:The effects of different geometric configurations (including a single cylinder and different arrangements of two coupled cylinders with uneven diameters) on the flow characteristics shall be investigated.
Research questions: What are the key parameters governing the VIV in the near-wall configuration? What are the interactions between identified parameters? How suitable are tools used in the present study for practical design tasks concerning subsea pipelines?
1.3 Outline of the Thesis
Structure of the Thesis was established in the following way:
Chapter 2: Flow around Circular Cylinder and Fluid-Structure Interaction introduces the fundamental theory of viscous flows, classification of flow regimes with respect to Reynolds number and classification of vortex shedding modes. The concepts of tur- bulence, boundary layers and, flow separation are briefly treated. Vortex shedding, dynamic equations of motion and vortex-induced vibrations theories are reviewed.
Chapter 3: Computational Fluid Dynamics presents general concepts of CFD and Finite Volume Method. Fundamental governing equations and numerical method used are explained. Description of the OPENFOAM code used for the simulations is provided together with a short discussion on the selected solution methods.
Chapter 4: Two degree-of-freedom near wall VIV in laminar vortex street regime de- scribes the process of a model creation and set up used in the numerical simulations of 2DoF cylinder near a horizontal plane wall at Re = 200.
Chapter 5: Vortex-induced vibrations of two rigid cylinders with uneven diameters near the horizontal plane wall at low Reynolds number. This chapter contains the pa- per draft which will be submitted to the Journal of Fluids and Structures. Presented study investigates the effects of different configurations of rigidly coupled cylinders with uneven diameters on VIV in the vicinity of a plane wall at Re = 200.
Chapter 6: Two degree-of-freedom near wall VIV in upper transition regime de- scribes the convergence studies and validation of a numerical model based on URANS approach to study the VIV at Re =3.6×106.
Chapter 7: Conclusionsand recommendations for future work.
1.4 Previous Work
The effect of vortex-induced vibration was known since the ancient times when it was first observed that wind can excite a taut wire of an Aeolian harp. The work of Henri Bé- nard and Theodore von Kármán in the beginning of 20th century resulted in a discovery of the vortex street formation and it’s relation to the periodicity of the cylinder’s wake.
The wake of a cylindrical structure exhibits a large variety of complex phenomena stem- ming from the diverse instabilities in the transition regions. Flow around cylinders is well researched area of fluid dynamics and is covered by many comprehensive positions in the literature such as: Sarpkaya (2010), Sumer and Fredsøe (2006), Zdravkovich (1997). Extensive review on the flow induced vibrations was given in Blevins (1990) and Nakamura (2016).
Experimental Studies. The effect of a plane wall on a flow around horizontally placed cylinders was investigated experimentally by numerous authors. Most existing studies have focused on the transverse VIV of the cylinder with one degree of freedom due to the larger amplitude in the transverse direction than that in the streamwise direction. One of the first published studies were those by Feng (1968) and Anand and Tørum (1985).
PREVIOUS WORK 5
In his work Feng (1968) investigated cylinder’s vibrations in air flow while Anand and Tørum (1985) focused on the behavior of the cylinder in the water flow. In both cases, Feng (1968) and Anand and Tørum (1985) demonstrated the vortex shedding frequency lock-in. Experiments conducted by Bearman and Zdravkovich (1978) investigated the effect of gap ratio (e/D where e is the gap distance and D is the cylinder diameter) on the vortex shedding in the Reynolds number regime between Re = 2.5×104 and Re = 4.8×104. Their results indicate that for a stationary cylinder vortex shedding is suppressed if e/D < 0.3 thus vibrations cease at low gap ratios. Tsahalis and Jones (1981) performed model testing in a wave tank, tracking the response of the center of the pipe with an optical tracking system. It was observed that the proximity of the boundary plane reduced the maximum amplitude of vibration and the onset of VIV was shifted to higher velocities than without the boundary presence. Fredsøe et al. (1987) investigated the cross-flow vibration of cylinders near a rigid wall. They concluded that for the range of reduced velocities3 < Ur < 8and 0 < e/D < 1 the transverse vibrating frequency is noticeably larger than the frequency of vortex shedding from a stationary cylinder.
More recently Yang et al. (2009) measured the vortex shedding frequency and mode by the method of hot film velocimetry and hydrogen bubbles. Researchers investigated the influence of reduced velocity, mass ratio, gap ratio and stability parameter on the am- plitude and frequency of the cylinder vibrations. The results from the parametric study were summarized as follows: with decreasing mass ratio, the width of the lock-in ranges in terms ofUr and the frequency ratio (f /fn ) become larger; with increasing gap-to- diameter ratio(e/D), the amplitude ratio(A/D)gets larger but frequency ratio(f /fn) has a slight variation for the case of larger values ofe/D. Wang et al. (2013) investigated flow around a neutrally buoyant cylinder with a mass ratiom∗ = 0.1and a low damping ratioζ = 0.0173. Experiment covered range of Reynolds number3×104 ≤Re≤1.3×104 and reduced velocity 1.53 ≤ Ur ≤ 6.6. In contrast to the case of a stationary cylinder where vortex shedding was suppressed at a gap ratioe/D <0.3the elastically mounted cylinder was found to vibrate even at the smallest gap ratioe/D= 0.05. In the study by Hsieh et al. (2016), the velocity field was measured using a high-resolution particle im- age velocimetry (PIV) system. Vibration amplitude and oscillation frequency for different Ur, mean velocity field, turbulence characteristics, vortex behavior, gap flow velocity, and normal/shear stresses on the boundary were measured. The particle image velocimetry was also used by He et al. (2017) to investigate the vibrating cylinder at Re = 1072 and gap ratio rangee/D= (0−3.0). In both studies, the flow statistics and vortex dynamics revealed a strong dependence on the gap ratio. When the cylinder approaches the wall (e/D <2.0) the wake becomes more asymmetric and a boundary layer separation occur on the wall downstream of the cylinder. The critical gap ratio was identified to be at
aboute/D = 0.25, below that value the vortex shedding is irregular. For lower gap ratio is was found that only the upper shear layer can shed vortices.
Numerical Studies. A rapid development of numerical methods and increasing compu- tational power led to a dynamic growth of the number of published numerical studies.
Following review covers some recent, relevant developments in the field. The near-wall cylinder in the turbulent regime was subject of numerical investigations by Ong et al.
(2010) atRe= 3.6×106. The k−model was used to study the effects of a gap to diam- eter ratio, Reynolds number, seabed roughness and boundary layer thicknessδ. One of the findings of this study was that the drag coefficient (CD) increases ase/D increases for small e/D, reaching a maximum value before decreasing to approach a constant value. The mean pressure coefficient (Cp) around the cylinder was studied and devel- opment of an asymmetry for small gap ratio (e/D = 0.1) was reported resulting in net positive upward lift. The effect disappears for large gap ratios and Cp becomes sym- metric. Rao et al. (2013) investigated numerically the flow past a stationary cylinder at different heights above a no-slip plane ranging from very small gap casee/D = 0.005to freestream condition. They identified critical Re in each of the simulated cases where the transition from steady two-dimensional flow to three-dimensional flow occurs. The behavior of the vortical wake created by a cylinder placed in the boundary layer flow was studied using URANS approach by Harichandan and Roy (2012). Two dimensional model was resolved using FVM solver at Re = 100 and Re = 200 respectively. Au- thors of the study describe the "vortex-wrapping" phenomena as the shear layer from the top and bottom cylinder surface curl up destabilizing the shear layer on the plane wall downstream of the cylinder. The vortex shedding frequency for wall proximity flows was found to be higher than for the unconfined flows. The flow past tandem of cylinders placed near a plane wall was investigated by Tang et al. (2015). Numerical simulations using two-dimensional Navier-Stokes equations were solved with a three-step finite ele- ment method at a low Reynolds number ofRe= 200. Various gap ratios (e/D) and pitch lengths (L/Dwhere Lis the distance between the cylinder’s centers) were considered.
The mean drag coefficient of the upstream cylinder was found to be larger than that of the downstream cylinder for the same combination ofe/D andL/D. They noticed that change in the vortex shedding modes led to a significant increase in the RMS values of drag and lift coefficients. D’Souza et al. (2016) studied the dynamics of the flow of two cylinders along a moving plane wall atRe= 200. The interaction with the wall bound- ary layer was thus removed and the focus of the study was on the influence of the wall proximity effects and the wake dynamics. The study revealed an early transition from the reattachment to co-shedding behavior. Specifically atRe = 200for e/D = 0.5 the
PREVIOUS WORK 7
combined wake interference and wall proximity effects lead to a parallel double-row of vortices for the tandem cylinders. Turbulent flow dynamics was the main objective of the study of the flow around one cylinder (Abrahamsen Prsi˘c et al. (2016)) and two cylin- ders (Li et al. (2018)) in the vicinity of a horizontal plane wall. In both studies, Large Eddy Simulations (LES) with Smagorinsky subgrid scale model was used to accurately capture the turbulence effect atRe = 13.100. Abrahamsen Prsi˘c et al. (2016) investi- gated the effect of the incoming boundary layer profile by comparing the uniform inlet flow profile with two logarithmic boundary layer profilesδ = 0.48Dandδ = 1.6D. The importance of boundary layer thickness is manifested by decreasing RMS lift coefficient as the cylinder becomes immersed in the boundary layer. In Li et al. (2018) six sets of simulations were performed ate/D= 0.1,0.3and 0.5 and pitch lengthL/D= 2and 5.
The wall proximity demonstrated a decreasing effect on the mean drag coefficient of the upstream cylinder. The effects of the cylinder pitch length were manifested in the for- mation of cavity-like flow between the cylinders, with re-circulation zone being the most pronounced atL/D= 2ande/D = 0.1. Both studies conclude that 3D LES simulations with subgrid scale modeling offer clear improvements over the 2D RANS approach, more accurately capturing details of the turbulent flow and associated forces.
Relatively small number of studies focuses on the freely vibrating cylinder in a proximity of a horizontal wall. In Tham et al. (2015) a 2-DoF cylinder close to the plane wall was simulated using Petrov-Galerkin FEM formulation. Gap ratio ranging from 0.5Dto 10D and reduced velocitiesUr from 2 to 10 were analyzed atRe= 100. Tham et al. (2015) explored the origin of the increased streamwise oscillation of a freely vibrating cylin- der near-wall as compared to the isolated case. For gap ratios lower than e/D = 0.60 additional branch in amplitude response was identified, namely upper branch in addi- tion to the initial and lower branches. The effect of enhanced streamwise oscillation was explained based on the phase difference curves revealing positive net power trans- fer for gap ratios lower than e/D = 0.9. Study of Chung (2016) focused on a 1-DoF vibrating cylinder with low mass ratio and zero damping. The outcome of the numerical study atRe = 100 showed that cylinder vibration in the lock-in zone is controlled by either the Strouhal frequency or the natural frequency of the structure in a fluid. Strong dependence on the gap ratio and reduced velocity was experienced. The case of an im- pact with the wall was also investigated and has been proved to cause no change in the amplitude and frequency of cylinder vibration. A comprehensive study of the vortex- induced vibrations of a single cylinder freely vibrating in the proximity of the plane wall was presented in Li et al. (2016). Both 2D and 3D simulations were performed using a Petrov-Galerkin finite element formulation. Wall proximity effects were studied in the laminar flow regime atRe= 200in a series of 2D simulations. Reasons for the enhanced
streamwise oscillations were explained in detail. The wall proximity effect was revealed to contribute to the reduction of streamwise vibration frequency by half. Streamwise frequency lock-in combined with positive net energy transfer was identified as the main mechanism behind large streamwise oscillation amplitude. The 2D simulations were supplemented by 3D simulations atRe= 1000aimed at capturing the three-dimensional effects and assessing accuracy and validity of the 2D results. Over predictions of force coefficients in 2D simulations were revealed when the Reynolds number is higher than Re= 200.
1.5 Summary
The literature review presents the broad overview of the previous and current research focused on the VIV of elastically mounted cylinders. Large base of experimental and numerical work exists for the case of the freestream flow. Relatively small number of published research is focused on the near-wall effect in conjunction with the 2-DoF VIV.
Furthermore, most of the published studies are investigating moderate to high Reynolds number flows. The validity of 2D simulations for low Reynolds number flows is con- firmed in the available literature, on the other hand, the applicability of 2D URANS at very high Reynolds numbers has been explored only to a limited degree. It appears that further investigating the effects of wall proximity on vortex shedding mechanisms using numerical simulations at low Reynolds number could offer additional insight beyond the available literature. In the view of a very limited number of studies in the upper transi- tion regime additional study focused on very high Reynolds flow provides an opportunity for gaining better understanding of the VIV physics.
References
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Anand, N. M. and A. Tørum (1985). Free span vibrations of submarine pipelines in steady flows-effect of free-stream turbulence on mean drag coefficients. Journal of Energy Resources Technology 107(4), 415 – 420.
Bearman, P. and M. Zdravkovich (1978). Flow around a circular cylinder near a plane boundary. Journal of Fluid Mechanics 89, 33 – 47.
Blevins, R. D. (1990). Flow-Induced Vibration. Van Nostrand Reinhold.
Chung, M.-H. (2016). Transverse vortex-induced vibration of spring-supported circular cylinder translating near a plane wall.European Journal of Mechanics - B/Fluids 55(Part 1), 88 – 103.
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D’Souza, J. E., R. K. Jaiman, and C. K. Mak (2016). Dynamics of tandem cylinders in the vicinity of a plane moving wall. Computers & Fluids 124(Supplement C), 117 – 135.
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Feng, C. C. (1968). The measurement of vortex induced effects in flow past stationary and oscillating circular and D-section cylinders. Ph. D. thesis, University of British Columbia.
Fredsøe, J., B. Sumer, J. Andersen, and A. E. Hansen (1987). Transverse vibrations of a cylinder very close to a plane wall. Journal of Offshore Mechanics and Arctic Engineering 109, 52 – 60.
Harichandan, A. B. and A. Roy (2012). Numerical investigation of flow past single and tandem cylindrical bodies in the vicinity of a plane wall. Journal of Fluids and Struc- tures 33(Supplement C), 19 – 43.
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Hsieh, S.-C., Y. M. Low, and Y.-M. Chiew (2016). Flow characteristics around a circular cylinder subjected to vortex-induced vibration near a plane boundary.Journal of Fluids and Structures 65(Supplement C), 257 – 277.
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OUP Oxford.
CHAPTER 2
FLOW AROUND CIRCULAR CYLINDER AND VORTEX-INDUCED VIBRATION
The present chapter provides a theoretical background of viscous fluid flow around a circular cylinder and basic concepts of vortex-induced vibration.
2.1 Flow Around Immersed Cylinders
Bodies immersed in a fluid stream are characterized by viscous effects (shearandno-slip) occurring near the body surface and in the wake. Such flows can be classified as bound- ary layer flows. Boundary layer growth and flow separation are defined by fluid forces on a microscopic level (Blevins, 1990). The flow around the circular cylinder disturbed by its presence can be divided into four regions proposed by Zdravkovich (1997), as shown in Fig. 2.1:
1. Region of retarded flow. Local time-averaged velocity, u, in this region is smaller than the freestream velocityU∞.
13
2. Boundary layer attached to the surface of the cylinder. The thickness of the boundary layers δ is very small compared with diameter D, therefore high-velocity gradient normal to the cylinder surface is present and shear stress effects are significant.
3. Two sideways regions where the flow is accelerated (u > U∞) and displaced.
4. Wide downstream region of separated flow: wake, whereu < U∞.
2
3 3
4 1
U
Figure 2.1: Regions of disturbed flow around circular cylinder, (Zdravkovich, 1997, p.4).
Reynolds number governs the ratio of inertial force to the viscous force and is expressed as:
Re= U D
ν (2.1)
where ν is the kinematic viscosity, U is the flow velocity and D is the characteristic diameter. As the Reynolds number increases from zero the flow characteristics change considerably. Sumer and Fredsøe (2006) gave a concise summary of the flow regimes of a circular cylinder in steady current. Their classification is presented in Table 2.1.
2.1.1 Flow Regimes
Flows with a very small Re < 5 are characterized by a lack of separation (creeping flows). At about 5 < Re < 40 separation occurs in the form of a pair of vortices be- hind the cylinder. IncreasingRefurther up leads to the phenomenon of vortex shedding which first appears at Re = 40. The vortices are shed alternating from each side of the cylinder with a frequency called vortex-shedding frequency, denotedfst. The vor- tex street formed behind the body remains laminar in the range 40 < Re < 200 and shedding can be treated as two-dimensional due to small variation in the spanwise di- rection (Sumer and Fredsøe, 2006). Three-dimensional effects become significant after the transition to turbulence occurs in the wake (200< Re <300). AtRe >300the flow is characterized by the completely turbulent wake and 3D effects become significant in
TURBULENCE 15
this regime. The range between 300 < Re < 3×105 is called the subcritical regime.
Boundary layer around the cylinder is laminar in this wide range of Reynolds numbers.
Lower transition regime covers approximately,3×105 < Re <3.5×105, and separation occurs in the cylinder boundary layer at one side of the cylinder leading to an asymmet- ric mean lift. The supercritical regime, 3.5×105 < Re < 1.5×106, is characterized by a turbulent boundary layer separation at both sides of the cylinder with transition point located between the stagnation point and separation point. In the upper transition regime,1.5×106 < Re < 4×106, fully turbulent transition develops at one side of the cylinder. After exceedingRe >4.5×106, the boundary layer around the cylinder is fully turbulent and such flow regime is called the transcritical regime.
2.2 Turbulence
According to Batchelor (2000) turbulent flow is a pattern of fluid motion characterized by chaotic changes in pressure and flow velocity. Turbulence has inherent features which can be listed as follows:
1. Turbulent flows are characteized by random velocity fluctuations with a wide range of length and time scales.
2. The large-scale eddying motions are strongly influenced by the geometry of the flow.
In other words the boundary conditions govern the transport and mixing within the flow. The behavior of the small-scale motions can be predicted by a rate at which they receive the energy from the large scale motions in a cascading way. The smallest lenght scales are affected by the viscosity of the fluid.
3. Random nature of the fluctuations requires statistical methods to analyze it.
4. Turbulent flows are characterized by a large Reynolds number in the sense that the inertia forces dominate the viscosity effects in the flow.
5. Turbulent flows are always dissipative. This means that they lose energy and decay.
Ultimately the smallest eddies dissipate into heat through the molecular viscosity.
6. Turbulent flows are characterized by enhanced diffusivity. Turbulent diffusion is much greater than that of a laminar flow (molecular diffusivity). The highly diffusive turbulence causes rapid mixing and increased rates of mass, momentum, and heat transfer.
7. Turbulent flow is highly vortical, meaning that it is rotational and characterized by high levels of fluctuating vorticity. The vorticity vector is defined as the curl of the
Table 2.1:Flow regimes around circular cylinder (Sumer and Fredsøe, 2006, p.2).
No separation Creeping flow Re < 5
Fixed pair of symmetric vortices 5 < Re < 40
Laminar vortex street 40 < Re < 200
Transition to turbulence in wake 200 < Re < 300
A
A
Wake completely turbulent
A: Laminar boundary layer separation 300 < Re <3.5×105
A
B
A: Laminar boundary layer separation B: Turbulent boundary layer separation Boundary layer still laminar
3×105< Re <3.5×105
B
B
B: Turbulent boundary layer separation Boundary layer partly laminar partly turbulent 3.5×105< Re <1.5×106
C
C: Boundary layer completely turbulent at one side 1.5×106< Re <4×106
C
C
C: Boundary layer completely turbulent at both sides Re >4×106
velocity vector:
~
ω=∇ ×U~ = ∂w
∂y −∂v
∂z,∂u
∂z − ∂w
∂x,∂v
∂x −∂u
∂y,
(2.2)
Vorticity is a measure of rotational effects, being equal to twice the local angular velocity of a fluid particle. Flow is irrotational if the vorticity is equal to zero.
BOUNDARY LAYER CONCEPT 17
8. Turbulence is inherently three-dimensional. The term two-dimensional turbulence is only used to describe the simplified case where the flow is restricted to two di- mensions. Vortex stretching is the phenomenon responsible for the continuous three dimensional deformation of the primary vortices.
9. Turbulence is a continuum phenomenon and is governed by the equations of fluid mechanics. Even the smallest turbulent length scales are much larger than the molecular length.
2.3 Boundary Layer Concept
2.3.1 Laminar Boundary Layer
In the range of applicability of continuum mechanics, the flow of a real fluid has two distinct characteristics. Namely, the no-slip condition meaning that, at a solid surface, the velocity of a fluid relative to the surface is zero and the condition of no discontinuity of velocity (Ward-Smith, 2005). Based on those assumptions it is possible to derive the velocity profile expression in the region near a solid surface as depicted in Fig. 2.2.
Equations for the laminar boundary layer were derived from Navier-Stokes equations by u = U∞
u y = δ
u = 0.99U∞
Figure 2.2:Velocity profile in a boundary layer (Schetz and Bowersox, 2011, p.3).
Prandtl (1904) and for a steady, two-dimensional flow they take form (White, 2006):
∂u
∂x +∂v
∂y = 0 (2.3)
u∂u
∂x+v∂u
∂y =UdU
dx +ν∂2u
∂y2 (2.4)
where U = u(x,∞) is the freestream velocity, u is the horizontal velocity component andv is vertical the velocity component. Solution to this system of parabolic partial dif- ferential equations can be obtained numerically. In case of a laminar flow past a plate,
analytical solution was found by Blasius (1908) by using a similarity transformation.
Thickness of a boundary layer was customarily taken as the distance from the solid sur- face at which the velocity reaches the 99% of velocity in the main stream U∞. Blasius solution can be expressed in terms of the boundary layer thickness (White, 2011):
δ99%≈ 5.0x
√Rex (2.5)
where δ is the boundary layer thickness, x is the distance along the plate andRex = ρU x/µis the local Reynolds number. Blasius expression was derived and holds true in the situation with zero pressure gradient. Occurrence of flow separation and effects of pressure gradient are discussed in the next section.
2.3.1.1 Separation Point. Adverse pressure gradient effect is the main reason behind the separation of the flow. The momentum loss near the wall in the boundary layer of a flow moving against increasing pressure gradient leads to the backflow at the wall.
Bernouilli equation links pressure gradient to the edge velocityU∞(x), so for an increas- ing pressure the velocity decreases:
udu ds =−1
ρ dp
ds (2.6)
If the pressure varies in the direction of a flow it can greatly affect the behavior of the fluid. In case of a flow over the curved surface with curvature large compared to the boundary layer thickness the velocity profile forms as shown in Fig. 2.3. Second
Separation point
Increasing pressure, decreasing U∞
Separation bubble
∂u∂y =0
U∞ U∞
U∞
U∞
Figure 2.3: Separation of a boundary layer over curved surface (Schetz and Bowersox, 2011, p.24).
derivative of velocityuat the wall can be used to explain the flow separation. From the momentum equation (Eq. 2.4) at the wall, whereu=v= 0, we obtain:
∂τ
∂y
wall
= µ∂2u
∂y2
wall
=−ρUdU dx = dp
dx (2.7)
BOUNDARY LAYER CONCEPT 19
To ensure smooth transition with the freestream flow the velocity profile must have a negative curvature. The wall curvature has the sign of the pressure gradient. In case of an adverse pressure gradient the second derivative of velocity is positive at the wall but it must be negative at the outer layer (y=δ) to merge with the freestream flowU∞(x).
It is therefore required that point of inflection exists at which the curvature changes signs from positive to negative. It should be noted that laminar flows separates easily in adverse gradients whereas turbulent boundary layer is more resistant due to increased wall friction and heat transfer.
2.3.2 Turbulent Boundary Layer
Situation of a flow over a flat plate with a uniform velocity profile is presented in Figure 2.4. The formation of a boundary layer can be separated into three distinct regions. The laminar region, the transition region and, the turbulent region. In a close proximity to the plate surface, the fluid particles are subject to no-slip condition and their velocity becomes zero resulting in the formation of a viscous sublayer. This results in the shear stress development in the thin fluid layer close to a plate. After some distance the flow becomes unstable and eddies are formed. This region is called a transition region. It should be noted that transition process is unsteady and is difficult to predict, even with modern CFD codes (Çengel and Cimbala, 2010, p. 558). Further downstream the plate, a turbulent boundary layer is formed. The viscous sublayer is characterized by very high
Figure 2.4: Formation of the laminar and turbulent boundary layer (Çengel and Cim- bala, 2010, p.558).
turbulent energy production rate. It is estimated that in this region more than 30% of the total production and dissipation takes place (Cebeci and Cousteix, 2009). The turbulent boundary layer can be treated as a composite layer which is composed of an inner and outer region (Fig. 2.4). The inner layer is about the laminar boundary layer thickness and is further divided into a linear sublayer, buffer layer and, log-law region. The outer layer is about 85% of the total boundary layer thickness (Cebeci and Cousteix, 2009).
2.3.3 Plus Units
Plus units are non-dimensional variables used in turbulent boundary layer analysis. They are defined as:
y+= uτr
ν , U+= U
uτ, Reτ = uτH
ν (2.8)
where y+ is the non-dimensional distance from the wall, U+ is the non-dimensional velocity andReτ is the Reynolds number based on shear velocityuτ =p
τw/ρ,τw is the shear stress at the wall andH is the radius of the control volume.
2.3.4 Law-of-the-wall
The velocity profile of a fully turbulent boundary layer can be non-dimensionalized using the derived plus units. The curve shape in the inner layer close to the wall is common for all Reynolds numbers. This relation was first observed and described by von Kármán in 1930. Figure 2.5 shows the plot of the velocity distribution in the inner layer. This universal curve is referred to as the law-of-the-wall. The equations relating the velocity to the nondimensional distance from the wall can be derived for the linear sublayer and for the outer layer. Close to the wall the velocity varies linearly with the nondimensional wall distance:
U+=y+ (2.9)
Further away from the wall at a distance where the kinematic viscosity is negligible the velocity can be approximated by:
U+ = 1
κln(y+) +C+ (2.10)
where κ is the von Kármán constant, usually equal to 0.41, C+ is an integration con- stant, commonly taken as 5.1. Assuming the linear variation ofτw with increasingy, the velocity distribution in the inner layer is a function of the wall shear stressτw, kinematic viscosityν and the distance from the wally. The approximate range of the viscous sub- layer is in the range of0< y+ <5. The log-law region is defined at30 < y+ where Eq.
2.9 is a good approximation of the velocity profile. A buffer region exists in the range of5 < y+ <30where neither Eq. 2.10 nor Eq. 2.9 applies. This region is of particular importance in CFD simulations of turbulent flows using the wall function approach.
VORTEX SHEDDING 21
10-1 100 101 102 103
y+ 0
5 10 15 20 25
U+
Linear sublayer U + = y+ Logarithmic overlap Spalding‘s law of the wall
Figure 2.5:The inner layer of a turbulent boundary layer.
2.4 Vortex Shedding
Vortex shedding is a result of the basic instability which exists between the two free shear layers released from the separation points at each side of the cylinder into the downstream flow from the separation points. These free shear layers roll-up and feed vorticity and circulation into large discrete vortices which form alternately on opposite sides of the cylinder.
2.4.1 Mechanism of Vortex Shedding
Phenomenon of vortex-shedding is common for all flow regimes for Re > 40. Due to the reasons explained in the previous section the boundary layer over the cylinder surface will separate and shear layers will be formed. The separated boundary layer is characterized by high vorticity which is then subject of the shear layer roll up into the vortex with an identical sign. Vortices rotating in opposite direction are thus formed at both sides of the cylinder. These vortices are very sensitive to the disturbances in the flow and in consequence one of the formed vortices will become dominating and will draw the opposite vortex across the wake (Sumer and Fredsøe, 2006). The opposite sign of the vorticity carried by the drawn vortex leads to cut off of the dominating vortex from the boundary layer and formation of the free vortex. After the vortex is shed, it is convected downstream in the wake. The drawn vortex will now become dominant and will draw newly formed vortex on the opposite side. The whole cycle repeats on the opposite side of the cylinder. This leads to the cyclic shedding of the vortices behind the cylinder.
A B
C A
B
I) II)
Figure 2.6:Vortex shedding mechanism (Sumer and Fredsøe, 2006, p.8).
2.4.2 Vortex Shedding Patterns
The vortex shedding can be characterized by the pattern in which the vortices are formed.
A thorough review of the shedding patterns and proposed classification was given in Williamson and Roshko (1988), Williamson and Govardhan (2004), Jauvtis and Williamson (2004) and Morse and Williamson (2009). The symbols, patterns and corresponding ex- amples of vorticity contours were given in Table 2.2. Short classification of the modes based on the text by Williamson and Govardhan (2004) can be summarized as follows:
2S mode is characterized by the alternate shedding of one vortex from each side of the cylinder.
2P mode is distinguished by pairs of vortices of the same vorticity sign thus two vortices are shed from each side during one cycle.
P + S mode is formed by a pattern wherein each cycle a vortex pair and a single vortex are formed.
2Po mode is similar to 2P mode, but a secondary vortex in each pair is significantly smaller than the primary vortex.
2P + 2S mode is comprised of two vortex pairs, formed alternatively like in 2P mode but additional vortex appears in between.
2T mode can be recognized by a characteristic triplet of vortices formed in each half cycle. In each triplet, two vortices have the same sign and one vortex of opposite sign is formed.
2C mode is similar to 2T mode but a doublet of vortices is formed instead of a triplet.
Both vortices have the same sign and full cycle comprises formation of four vortices, two on each side of the cylinder.
HYDRODYNAMIC FORCES 23
Table 2.2:Vortex shedding patterns (Williamson and Govardhan, 2004).
2S
S S
2P
P P
P+S
P S
2Po
P P
2P+2S
P
S P
S
2T
T T
2C
C C
2.5 Hydrodynamic Forces
2.5.1 Forces on a Cylinder in Steady Current
Following discussion is based on Sumer and Fredsøe (2006). The total force acting on the surface of the cylinder in steady flow can be divided into two components, namely the pressure component and the viscous component. The forces acting in two directions:
in-line (IL) and cross-flow (CF) can be found by integrating the orthogonal components of pressure and viscous forces along the cylinder surface. The total force acting in the IL direction is the mean dragFD and is a sum of mean form dragFp (due to the pressure) and mean friction dragFf (due to the viscous forces). Force acting in the CF direction - mean lift forceFL- is defined in a similar way. In the case of a cylinder in the free stream flow, theFL will be equal zero due to the symmetry of the flow, however, forRe > 40 the vortex shedding phenomenon occurs and instantaneous CF force on the cylinder will be non-zero.
Change of the pressure distribution over the cylinder surface during a shedding cycle was shown in Fig. 2.7. It is evident that due to the periodic change of the pressure distribution the force components will also exhibit the periodic variation. Due to vortex
U
F
F
F
F
Figure 2.7: Development of pressure distribution and force during vortex shedding cycle (Sumer and Fredsøe, 2006, p.38).
shedding the velocity of the flow around the cylinder will change periodically at the top and bottom surface. When a vortex is shed at the lower edge of the cylinder the upward lift is developed. The velocity will be higher on the lower edge and the pressure at the bottom will then be larger than at the top according to Bernoulli’s equation. This induces the force which will be directed upwards. After the other vortex is developed at the opposite side, the situation is inverse and result is a force directed downwards.
Each complete vortex shedding cycle comprises the effects of a complementary pair of vortices. Cylinder with low structural damping properties will be thus forced to vibrate in the CF and to a lesser extent in the IL directions.
2.5.2 Drag and Lift Coefficients
The dimensionless drag coefficient (CD) for a smooth cylinder is a function ofReand is expressed as:
CD = FD
1
2ρU2A (2.11)
The lift coefficient is defined as:
CL= FL 1
2ρU2A (2.12)
DYNAMIC EQUATIONS OF MOTION 25
where: FLis the mean lift force,FD is the mean drag force,ρis the density of a fluid,U is the flow velocity andAis the projected area perpendicular to the incoming flow. The denominator in both expressions is the product of the dynamic pressure of the undis- turbed flow and the specified projected area. Therefore as a ratio of two forces the drag and lift coefficients are the same for two dynamically similar flows.
2.5.3 Pressure Coefficient and Skin-friction Coefficient
The pressure coefficient gives the ratio of static pressure to dynamic pressure and is expressed as:
Cp = p−p∞ 1
2ρU∞2 (2.13)
where p is the static pressure at the evaluated point, p∞ is the static pressure in the freestream,ρis the density of a fluid andU∞is the freestream flow velocity.Cpequal one indicates stagnation point as it corresponds to stagnation pressure. The dimensionless number that relates the wall shear stress to dynamic pressure is skin-friction coefficient, defined as:
Cf = τw
1
2ρU∞2 (2.14)
whereτw =µ∂u∂y is the local wall shear stress. An approximate point of separation can be found at the location whereCf = 0indicating thatτw changes sign from positive to negative.
2.6 Dynamic Equations of Motion
2.6.1 Solutions of a Viscous Damped Vibration Equation
In case of a single DoF freely oscillating system the equation of motion in the form (Inman, 2013):
mx(t) +¨ cx(t) +˙ kx(t) = 0 (2.15) wheremis the mass of the system,cis the damping coefficient,kis the spring stiffness, x(t) is the displacement, x(t)˙ denotes the velocity andx(t)¨ is the acceleration. To find the solution to Eq. (2.15) one can write the equation in the form:
(mλ2+cλ+k)aeλt= 0 (2.16)
